Existence of positive solutions of elliptic equations with Hardy term

Abstract

This paper is devoted to studying the existence of positive solutions of the problem: \begin{equation} \begin{cases}\label{0.1}\tag{$\ast$} -\Delta u=\frac{u^{p}}{|x|^{a}}+h(x,u,\nabla u), & \mbox{in} \ \Omega,\\ u=0, & \mbox{on}\ \partial\Omega,\\ \end{cases} \end{equation} where $\Omega\subset \mathbb{R}^{N}(N\geq3)$ is an open bounded smooth domain with boundary $\partial\Omega$, and $1<p<\frac{N-a}{N-2}$, $0<a<2$. Under suitable conditions of $h(x,u,\nabla u)$, we get a priori estimates for the positive solutions of problem \eqref{0.1}. By making use of these estimates and topological degree theory, we further obtain some existence results for the positive solutions of problem \eqref{0.1} when $1<p<\frac{N-a}{N-2}$.

Keywords

• a priori estimates
• hardy term
• positive solutions